R isk Modeling of Multi-year , Multi-line Reinsurance Using Copulas

1. Risk Modeling of Multi-year, Multi-line Reinsurance Using Copulas by Ping Wang St John’s University, New York on CICIRM 2011 at Beijing, China

2. Agenda Today • Multi-year, multi-line reinsurance • A Framework Using Copulas to model time dependence • Application using real data • Concluding remarks • Q & A

3. Multi-year, multi-linereinsurance policies • Cover losses arising from multiple lines of business over multiple years (3 or 5 most common) • Stop-loss type, commonly. Reinsurer pays claims only if the accumulated losses from several business lines over an extended period exceed a fairly high threshold. • Reduced volatility compared to separate coverage

4. Difficulty Facing Actuaries • Simultaneous modeling dependence • Across time, and • Across business lines (e.g., workers compensation and commercial multiple perils)

5. Modeling Product Risk With Copula • Assume independence between business lines • Model time-dependence of each line using copula • Simulate the distribution of future accumulated losses • Estimate the payoff of multi-year, multi-line reinsurance

6. Marginal Distribution • Suppose that there are Ti years data for a business line of the ith primary insurer • Univariate marginal distribution functions • Fit with Gamma, normal, lognormal, t-dist’n

7. Modeling Time Dependencies Using Copulas • With Copula C, the joint distribution function of Yi can be expressed as • The log-likelihood of ith primary insurer is • where c(.)is the probability density function corresponding to the copula function • Predictive distribution is obtained based on the results of maximum likelihood estimation

8. Estimate Product Risk • Simulation of joint distribution of each business line over multiple years • Calculate the policy payoff • Analyze the risk using VaR and CTE

9. Real Data • Loss ratios of workers compensation (WC) and commercial multiple perils (CMP) • 32 primary insurers • Task: based on the loss history of 5 years, fit the multivariate distribution, simulate the future losses, then model the risk of the reinsurance policy that covers accumulated losses of both lines over next three years.

10. Correlations across Time: WC • Loss ratios among years are not independent.

11. Correlations across Time: CMP

12. Relationship between WC & CMP • Correlation coefficient: 0.1510

13. Fitted Marginal Distribution *: kolmogorov-Smirnov test statistic

14. t-copula • t-copula: • where Gr is CDF of t-distribution function and

15. Different “correlation matrices”

16. Maximum Likelihood Estimation • Parameters to be estimated: • of copula:  in correlation matrix Σ and degrees of freedom r • of marginal distribution, e.g. shape and scale parameters for Gamma

17. MLE Results: WC

18. MLE Results: CMP

19. Simulation and Analysis • Based on the multivariate distribution of the loss ratio for business lines (WC, CMP separately) for the primary insurer • Simulate the multivariate variables and • The overall loss across two lines over three years is • Where P denotes the annual premium • Payment on the reinsurance policy after deductible D

20. Histogram of Total Loss Using Different Assumptions

21. VaR and CTE of Total Loss (in millions)Using Different Assumptions • Of 10,000 simulations of Total Loss • Based on temporal independent loss ratios 196 are greater than the threshold; the reinsurer expects claims at a frequency of one in about fifty years, with average claims of $24.50 million. • Based on copula dependence the frequency of claims is about 5% (495 of 10,000), or one in twenty years, and the average claims $41.71 million.

22. Remarks • Copulas • can use information developed over time to better fit the multi-year claims experience • Can use information from similar risk classes

23. Questions and comments? Thank You!